We investigate how very large populations are able to reach a global consensus, out of local “microscopic” interaction rules, in the framework of a recently introduced class of models of semiotic dynamics, the so-called naming game. We compare in particular the convergence mechanism for interacting agents embedded in a low-dimensional lattice with respect to the mean-field case. We highlight that in low dimensions consensus is reached through a coarsening process that requires less cognitive effort of the agents, with respect to the mean-field case, but takes longer to complete. In one dimension, the dynamics of the boundaries is mapped onto a truncated Markov process from which we analytically computed the diffusion coefficient. More generally we show that the convergence process requires a memory per agent scaling as $N$ and lasts a time $N^{1+2∕d}$ in dimension $d⩽4$ (the upper critical dimension), while in mean field both memory and time scale as $N^{3∕2}$, for a population of $N$ agents. We present analytical and numerical evidence supporting this picture.

• Received 23 September 2005

DOI:https://doi.org/10.1103/PhysRevE.73.015102